Kinetic and related models with applications in the natural sciences

A brief overview of why and when the wave turbulence closure works.

Alan Newell

University of Arizona

Since the 30's, turbulence theories have wrestled without much success with the problem of closure, replacing an infinite hierarchy of moment or cumulant equations with a finite set. Because of weak nonlinearity and separation of time scales, wave turbulence, the study of the long time statistical behavior of a sea of weakly nonlinear dispersive waves, has a natural asymptotic closure. All properties can be calculated in terms of a single (or set of) two point function (resp. functions) and it itself satisfies a single (or set of) closed equation (resp. equations) called the kinetic equation. The derivation of the closure depends on certain premises the validity of which I will discuss. There are surprises! I will also discuss the ranges of validity in wavenumber space for the finite flux Kolmogorov-Zakharov (KZ) solutions and indicate what happens outside of these ranges. For those unfamiliar with the derivation of the kinetic equation, the organizers have arranged a separate time for me to present this calculation in the simplest of contexts.